This problem comes from a exercise of Rings and Categories of Modules.
Let $M$ be a left $R$-module, proof that every epimorphism $f:M \rightarrow R$ splits.
(If $f:M \rightarrow N $ and $f':N \rightarrow M$ are homomorphisms with $ff'$= $1_N$, we say that $f$ is a split epimorphism.)
My first idea is :
For $\forall r\in R$, I choose $m\in M$ which satifies $f\left( m \right) =r$.
Then I define $$f':R\rightarrow M$$ $$f'(r)=m$$.
I would like to know if the idea is correct? Perhaps I do not think well.
From the Comment section, I get the following picture.
$R$ is a free $R$-module, so $R$ is a projection module, so exists $f':R\rightarrow M$ which satisfies $ff'=1_R$.
THanks!
